After a little linear change of variables Solve[{d1 == \[Alpha]3 - \[Alpha]2, d2 == \[Tau]1 + \[Tau]2 + \[Alpha]4}, {\[Alpha]3, \[Tau]1}] you can rewrite your expression so that all signs become +:

myExpression = 
  2 (qx^2 + qz^2) (B1 (Ka1 + Kn1) qx^4 + Ka1 Kn1 qx^6 + 
      B0 (B1 + 
         Kn1 qx^2) qz^2) (\[Alpha]2 - \[Alpha]3) \[Rho] (qx^4 \
\[Lambda]p (\[Alpha]4 + \[Tau]2) + 
      qz^4 \[Lambda]p (\[Alpha]4 + \[Tau]2) + 
      2 qx^2 (1 + 
         qz^2 \[Lambda]p (\[Alpha]4 + \[Tau]1 + \[Tau]2))) + (-2 B1 \
qz^2 \[Rho] + 2 Ka1 qx^6 (\[Alpha]2 - \[Alpha]3) \[Lambda]p \[Rho] + 
      qz^4 (\[Alpha]2 - \[Alpha]3) (\[Alpha]4 + 
         2 B0 \[Lambda]p \[Rho] + \[Tau]2) + 
      qx^4 (-2 Kn1 \[Rho] + (\[Alpha]2 - \[Alpha]3) (\[Alpha]4 + 
            2 (B1 + Ka1 qz^2) \[Lambda]p \[Rho] + \[Tau]2)) + 
      2 qx^2 (-B1 \[Rho] + 
         qz^2 (-Kn1 \[Rho] + (\[Alpha]2 - \[Alpha]3) (\[Alpha]4 + (B0 \
+ B1) \[Lambda]p \[Rho] + \[Tau]1 + \[Tau]2)))) (2 B0 qx^2 qz^2 \
\[Alpha]2 - 2 B0 qx^2 qz^2 \[Alpha]3 - Kn1 qx^6 \[Alpha]4 - 
      2 Kn1 qx^4 qz^2 \[Alpha]4 - Kn1 qx^2 qz^4 \[Alpha]4 + 
      B0 qx^4 qz^2 \[Alpha]2 \[Alpha]4 \[Lambda]p + 
      2 B0 qx^2 qz^4 \[Alpha]2 \[Alpha]4 \[Lambda]p + 
      B0 qz^6 \[Alpha]2 \[Alpha]4 \[Lambda]p - 
      B0 qx^4 qz^2 \[Alpha]3 \[Alpha]4 \[Lambda]p - 
      2 B0 qx^2 qz^4 \[Alpha]3 \[Alpha]4 \[Lambda]p - 
      B0 qz^6 \[Alpha]3 \[Alpha]4 \[Lambda]p - 
      2 B0 Kn1 qx^4 qz^2 \[Lambda]p \[Rho] - 
      2 B0 Kn1 qx^2 qz^4 \[Lambda]p \[Rho] - 
      2 Kn1 qx^4 qz^2 \[Tau]1 + 
      2 B0 qx^2 qz^4 \[Alpha]2 \[Lambda]p \[Tau]1 - 
      2 B0 qx^2 qz^4 \[Alpha]3 \[Lambda]p \[Tau]1 - (qx^2 + 
         qz^2)^2 (Kn1 qx^2 + 
         B0 qz^2 (-\[Alpha]2 + \[Alpha]3) \[Lambda]p) \[Tau]2 + 
      Ka1 qx^4 (qz^4 (\[Alpha]2 - \[Alpha]3) \[Lambda]p (\[Alpha]4 + \
\[Tau]2) + 
         qx^4 \[Lambda]p (-2 Kn1 \[Rho] + (\[Alpha]2 - \[Alpha]3) (\
\[Alpha]4 + \[Tau]2)) - 
         2 qx^2 (\[Alpha]3 + 
            qz^2 \[Lambda]p (Kn1 \[Rho] + \[Alpha]3 (\[Alpha]4 + \
\[Tau]1 + \[Tau]2)) - \[Alpha]2 (1 + 
               qz^2 \[Lambda]p (\[Alpha]4 + \[Tau]1 + \[Tau]2)))) + 
      B1 (-qz^4 (\[Alpha]4 + 2 B0 \[Lambda]p \[Rho] + \[Tau]2) + 
         qx^6 \[Lambda]p (-2 (Ka1 + 
               Kn1) \[Rho] + (\[Alpha]2 - \[Alpha]3) (\[Alpha]4 + \
\[Tau]2)) + 
         qx^2 qz^2 (\[Alpha]4 (-2 + 
               qz^2 (\[Alpha]2 - \[Alpha]3) \[Lambda]p) - 
            2 B0 \[Lambda]p \[Rho] + 
            qz^2 (\[Alpha]2 - \[Alpha]3) \[Lambda]p \[Tau]2 - 
            2 (\[Tau]1 + \[Tau]2)) - 
         qx^4 (2 \[Alpha]3 + \[Alpha]4 + \[Tau]2 + 
            2 (qz^2 \[Lambda]p ((Ka1 + 
                    Kn1) \[Rho] + \[Alpha]3 (\[Alpha]4 + \[Tau]1 + \
\[Tau]2)) - \[Alpha]2 (1 + 
                  qz^2 \[Lambda]p (\[Alpha]4 + \[Tau]1 + \[Tau]2))))));
Collect[myExpression /. 
  Solve[{d1 == \[Alpha]3 - \[Alpha]2, 
     d2 == \[Tau]1 + \[Tau]2 + \[Alpha]4}, {\[Alpha]3, \
\[Tau]1}][[1]], qx, Collect[#, qz, Simplify] &]

The expression will be positive whenever all variables are positive. I don't know if this is of much help to you.